# Collective modes in the paramagnetic phase of the Hubbard model

## TL;DR

This paper investigates the charge excitation spectra and collective modes of the Hubbard model in the paramagnetic phase, revealing a continuum, a gapless zero-sound mode, and a high-frequency mode, with detailed analysis of their dependence on interaction strength and temperature.

## Contribution

It provides a detailed analytical and numerical study of charge excitations and collective modes in the Hubbard model using slave-boson methods, including an approximate expression for the high-frequency mode.

## Key findings

- Identification of a gapless zero-sound mode with anisotropic velocity
- Derivation of an analytical expression for the high-frequency mode
- Finite temperature phase diagrams showing suppression of magnetic instabilities

## Abstract

The charge dynamical response function of the Hubbard model is investigated on the square lattice in the thermodynamical limit. The obtained charge excitation spectra consist of a continuum, a gapless collective mode with anisotropic zero-sound velocity, and a correlation induced high-frequency mode at $\omega\approx U$. The correlation function is calculated from Gaussian fluctuations around the paramagnetic saddle-point within the Kotliar and Ruckenstein slave-boson representation. Its dependence on the on-site Coulomb repulsion $U$ and density is studied in detail. An approximate analytical expression of the high frequency mode, that holds for any lattice with one atom in the unit cell, is derived. Comparison with numerical simulations, perturbation theory and the polarization potential theory is carried out. We also show that magnetic instabilities tend to vanish for $T\gtrsim t/6$, and finite temperature phase diagrams are established.